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Optimization Methods - Overview

Introduction​

Overview of optimization problems in linear algebra: minima, maxima, saddle points, and the finite element method.

Key Concepts​

Minima, Maxima, and Saddle Points​

  • Critical points: ∇f = 0
  • Second derivative test
  • Hessian matrix
  • Constrained optimization

Quadratic Functions​

  • f(x) = ½xᵀAx - bᵀx
  • Minimum when A is positive definite
  • Connection to least squares

Lagrange Multipliers​

  • Constrained optimization
  • KKT conditions

Finite Element Method​

  • Variational formulation
  • Weak form of PDEs
  • Stiffness matrix
  • Applications to engineering

Applications​

  • Machine learning optimization
  • Engineering design
  • Structural analysis
  • Computational physics

References​